The analysis of the eight classroom sessions resulted in the identification of 34 connections. Stage 1 of the analysis revealed that in a classroom context a connection can be a network of relations called links. Therefore, connections can be understood as semiotic functions that many times appear as chains or sets of other semiotic functions (the links). Each of the links is sustained by a relation between objects, such as the equivalence between representations or procedures, the application of procedures, justifications, generalisations, implications, the emphasis on a particular case or the existence of a common feature. Figure 2 shows the general structure of connections in the classroom. The numbering of the links is defined by their chronological appearance. The maximum \(n\) that we have identified is 6. Connections can be made up of one single link or many coordinated links.

Elementary structure of connections as a network of links

The results of the analysis led to the characterisation of the connections and the links that shaped their internal structure. Figure 3 shows the four categories of connections that were identified: intra-mathematical conceptual connections with treatment (IMCT) and with conversion (IMCC), intra-mathematical connections related to processes (IMCRP) and extra-mathematical connections (EMC). The connections have been numbered following their chronological order of appearance.

Classification of connections

Intra-mathematical conceptual connections are relations established between representations, definitions, operations, properties, procedures, justifications, and models associated with a concept. These connections are in turn differentiated into two categories, those that involve conversions and those that only entail that a treatment appears, in the sense proposed by Duval (2006).

The intra-mathematical conceptual connections with treatment make up most of the connections identified in the classroom. They are connections that are made between objects related to the same concept, without changing registers (Duval, 2006). During the analysis, 22 conceptual intra-mathematical connections with treatment were identified. Although the 22 connections entail a transformation between representations in the same register, there are some connections that emphasize the characteristics of the representations used; connections that emphasize definitions; connections that emphasize procedures; and connections that emphasize patterns and properties. These different emphases result in four sub-categories of intra-mathematical conceptual connections with treatment. In Table 2, the different sub-categories of IMCT are described, and the connections belonging to each sub-category are listed.

Although all connections are made explicit through particular representations, their classification depends on what the connection emphasises. For instance, in C21 (Table 3; Fig. 4), the difference between \(\frac^}^}\) when \(p\ge q\) and when \(p<q\) is discussed and the relation between representations is emphasised. Iván’s utterance (l1) what if p is smaller than q? connects the representations of the previous case (\(p\ge q\)) with a new possible representation without the previous restriction. Next, his own inference (would it be negative?) along with the teacher’s answer (l2) introduces a link between powers with natural exponents and powers with integer exponent. In C22, on the contrary, the equivalence of procedures for calculating \(\frac^}^}\) is discussed focusing on their suitability for performing the calculation, and the emphasis is placed on why the two procedures are equivalent.

Internal structure of the connection C21

One single intra-mathematical conceptual connection with conversion was identified, C15 (Table 4), which seeks to coordinate (l4) powers with integer bases with a real model (going to the basement –2 of a building). In this case, the logic of the model does not determine the interpretation of the operations (l3), since it does not seek to use a metaphor of the power in a real context, but the model is used to introduce a change in register which helps to show the difference between the order in which the two operations should be applied in each case (l1 and l2). Thus, it is not an extra-mathematical connection, but an intra-mathematical connection with conversion.

Intra-mathematical connections related to processes are relations established between a mathematical concept and a mathematical process that transverses the mathematical activity. More specifically, they are considered connections that establish relations with reasoning and justification, with the communication of mathematical information and with the heuristics related to problem-solving.

For example, connection C1 (Table 5; Fig. 5) emerges when solving an activity on the greatest common divisor and the lowest common multiple of a pair of numbers. A student observes that there is regularity when one of the numbers is a multiple of the other and intervenes in class to mention it. A connection is established between the calculation procedure of the l.c.m. and the g.c.d., and a property related to the l.c.m. and the g.c.d. of pairs of numbers, where one is the multiple of another. The establishment of this relation further explores the concepts of l.c.m. and g.c.d., since both the multiples and the denominators of both numbers are analysed exhaustively. The first link (l1) produced is between the general calculation method and the particular case in which one number is the multiple of the other (Table 5, Fig. 5). Below, the utterances of the students and the teacher highlight the property observed in the previous particular case, generalising for any pair of numbers in which one is a multiple of the other (l2). Based on the previous generalisation, the teacher establishes another generalisation formulating the previous property in a more elaborated way (l3). By demonstrating in a detailed and rigorous manner, the previous generalisation is further explored (l4). Lastly, through the final assessment of what it means to demonstrate, the teacher justifies the importance of the justification method that she has just shown (l5).

Internal structure of the connection C1

The establishment of this connection may help students to understand that to be sure that a mathematical property is fulfilled, a rigorous justification must be performed in all cases, and we must not limit ourselves to accepting an inductive result, which is fulfilled in some particular cases, meaning it is a connection related with processes.

To identify this category of connections, it is necessary to identify an explicit element which serves to interpret that the connection goes beyond considering a specific concept and focuses on a more general process in mathematics which is applied to a wide variety of concepts. For instance, in the previous example, the explicit utterance of the teacher that defines l5 is what represents the move from a specific situation (the proof that the pattern can be generalised) to a process (the difference between inductive and deductive reasoning).

Eight intra-mathematical connections related with processes were identified, among which three blocks are differentiated: those related with explanation, justification and demonstration, as in the previous case (C1, C5, C7, C17 and C27); those related with accuracy in the use of mathematical language (C33); and those related to the ability to decide based on why and when to use a procedure or a property (C20, C29).

Extra-mathematical connections are established between an extra-mathematical reference and a mathematical content. Two main types of extra-mathematical connections are defined. First, the connections in which extra-mathematical references are used to improve comprehension of the mathematical concepts, such as the basic metaphors (grounding) described by Lakoff and Núñez (2000). Second, the connections that show applications of mathematics in extra-mathematical situations, such as the knowledge of aspects of real life both in everyday terms and in other curricular disciplines such as visual art, social science, experimental science, economics, technology and music. Although we did not identify any connection of this second type, as there was no application of mathematics to any other context, it is relevant to mention them as they have been widely studied in mathematics education.

Connection C4 is an example of a connection in which extra-mathematical references are used to improve comprehension of the mathematical concepts. The connection is created between the addition and subtraction of integers and a model of movement which helps to interpret them. While the students solve \((-8)-(-4)\), a connection is established between and a model of movement in which there is a positive direction (whether to the right or up) and a negative direction (the opposite to the previous one, respectively). Below, Table 6 displays the explicit utterances that define each link of connection C4 and Fig. 6 shows its structure.

Internal structure of the connection C4

During the analysis, 3 extra-mathematical connections were identified (C4, C13, C31). Two of them refer to connections the teacher made between mathematical language and computer language (C13), and between mathematical language and musical language (C31). In both cases, characteristics of non-mathematical language were used to show the importance of understanding and using mathematical language correctly.

The previous characterisation of connections was obtained through a global analysis of connections that considered the role that each connection played in the construction of mathematical knowledge, in the classroom context where it was made. However, since most of the connections (27 out of 34) were formed by more than one link, it is necessary to consider how the links that form the connection are coordinated and how these links inform of the nature of mathematical connections.

Throughout the analysis, we identified 12 types (themes) of links that can be grouped in 4 categories: representational, based on common feature, procedural and argumentative (Fig. 7). Below, we present a detailed description of each category and provide examples of each type. The numbers in parentheses stand for the number of the link in the internal structure of the connection that precedes the parentheses. For instance, C12(2) stands for the link 2 in C12.

Classification of links at stage 3 and stage 4 of the analysis

Representational: The link is established between two representations of the same mathematical object (\(_\to _\)). The link can relate equivalent representations (ER), when both representations belong to the same register (e.g. in C12(2) the teacher links \(-^\) and \(-^\)) to emphasize the base of the power) or alternate representations (AR), when there is a change in the register (e.g. representing numbers geometrically).

Based on common features: The link is established between two objects \(_\to _\) that share a common feature, without being equivalent. These links are triggered by erroneous and ambiguous interpretations of mathematical objects by the students. The common features that sustain the link can be related to their definition (CFD), for instance in C18(2) when a student makes the mistake \(^=-1\cdot 36\) and refers to the power as a multiplication. The student identifies that the notion of multiplication is used in both operations and assigns an erroneous meaning to \(^\). This category of links may also be related to commonalities in the representation (CFR), for instance in C15(1,2,3,4) (Fig. 4) or in C34(1), when a student explicitly refers to \(\frac\) and \(\frac\) as being equivalent. In this case, the student uses the numbers represented in the fraction \(\frac\) and searches for a way to obtain an integer result, assigning an erroneous meaning to \(\frac\). Finally, they can be related to some metaphorical projections (MP) in which the metaphor is used in a literal way. For instance, in C4(2), when a student interprets \(-4-(-8)\) as a metaphor of being in basement 8 and going up 4 floors.

Procedural: This category refers to two different types of links. The first is called procedure link (P) and refers to links between a concept and a procedure (\(C\to P\)) that can be used when dealing with the concept. For instance, in C4(1), the teacher suggests that using the rules for operating with integers to obtain a notation without double symbols is a good resource to make the calculation. The second one is called equivalent procedures link (EP) and refers to links (\(_\to _\)) between two procedures that are useful for solving the same task. For instance, in C22(1,2,3,4,5,6), students propose four different procedures for solving \(\frac^}^}\) and the teacher conducts a discussion on their equivalence.

Argumentative: The link is established between two propositions (\(_\to _\)), where \(_\) stands for a premise and \(_\) stands for a conclusion. This category of links includes reasoning of several kinds. First, it refers to justifications (JU) of practices in mathematics. For instance, in C1(5) (Table5; Fig. 5), the teacher discusses the difference between inductive reasoning and deductive reasoning emphasizing the meaning of a mathematical proof; in C27(1), the teacher discusses the difference between \(\sqrt\) and \(\sqrt+\sqrt\) providing numerical examples; or in C21(2), the teacher justifies the use of the representation \(^, k\in N\). Second, it refers to arguments built upon a principle of transitivity (TR). For instance, in C16(2), a student asserts that two operations are the same because they have the same results and the teacher answers providing counterexamples of different operations that have the same result, explicitly emphasising that having the same result does not mean that the operations are the same. Third, it refers to implications (IM) or if–then arguments. For instance, in C14(3), when discussing the difference between \(-^\) and \(^\) the teacher shows three different representations of \(^\)(\(^;2\cdot 2\cdot 2\cdot 2\cdot 2;\) and \(32\)) and argues that if we change the sign of all three of them, then they would also represent the same operation. Fourth, it can refer to the application of a property to a particular case (PC). For instance, in C7(1), the teacher examines a particular case of a procedure previously proposed by a student for solving \(-4-(-8)\). Finally, it can refer to generalisations (GE) of a property to a broader set, as in C21(1), when a student proposes a generalization of \(\frac^}^}\) when \(p\ge q\) to the case when \(p<q\).

Table 7 shows the relation between the categories of connections and the categories of links that were identified during the analysis of the eight sessions. The reduced amount of extra-mathematical connections and intra-mathematical conceptual connections with conversion that were identified in the analysis makes it difficult to observe a relation between them and specific types of links. However, in the case of intra-mathematical connections related with processes, they are mostly related to argumentative links. In the case of intra-mathematical conceptual connection with treatment, all the types of links were identified, which signals the diversity of such a kind of connections. Other implications of the coordination between the global and specific analyses (Table 7) are discussed in the next section.